Classical stability analysis implies that a stable system –Eq. (REF )– will exhibit exponentially damped responses with time whereas a system that is unstable –Eq. ()– will exhibit exponential amplification. However, at the boundary between the two regimes –Eq. ()–, the method fails to accurately predict the long-time linear stability of the system, as shown in the previous work by King et al. [1]} and Huber et al. [2]}. Specifically, King et al. examined a one-dimensional (1D) operator (henceforth referred to as 1D-KRK)Notation is chosen to reflect the dimesionality of the operator and the lead author's initials that governs the response to varicose perturbations in a curtain flow, and Huber et al. studied a 1D operator (henceforth referred to as 1D-CMH)[] that enabled the neutral stability threshold in Eq. () to be traversed by the variation of a parameter. At the neutral stability boundary, neither operator should result in growing or decaying responses as per the classification given by Eq. (); however, both operators admit solutions that exhibit algebraic (i.e., \(t^p\) , \(p\)  real and rational) growth in time. In this paper, we examine the 2D extensions of the 1D-KRK and 1D-CMH operators, denoted respectively as the 2D-KRK and 2D-CMH operators.
